Today's opener primes my students to use function notation throughout the week. My students have seen function notation before, but more as a side-note than as a focal point to a lesson. This is by design. I consider function notation a tool that makes the most sense in context. If students can use function notation in a variety of ways, then they'll be more comfortable with it than they'd be if they just practiced a series of exercises without context. With that in mind, I want to move quickly through the decontextualized exercises that comprise this opener. For everyone who understands this task, it's a great warm-up, and anyone who doesn't get it will soon see what it means to apply this tool on today's assignment.

As students get started, I circulate to see how they're doing. At each group, I try to identify an expert who's got this down, and if I can't, I stick around to show kids how to work the first example. When I can see that everyone has at least given this a try, I begin to write the solutions on the board. I take the reins here because I want to neatly and consistently model what it looks like to input each value of x into the function rule. After the first example, I skip "showing" the intermediate step, because I want students to understand that "showing work" doesn't have to mean the same thing as "writing a lot".

To reiterate the idea that x is an input to a function rule and f(x) is the output for a given value of x, I also make a table of values off to the side of this work, and fill it in with the values from each of these four exercises. This move results in a satisfying "aha!" from a lot of kids, who hadn't yet really made sense of the notation.

Finally, I point to the last exercise, e, and ask for a volunteer to interpret what we're being asked to do here. There are several possibilities for what they might say, and the key is to listen for the mathematical truth and use it. For example, in one of my classes, a student was excited to point out that "it's the same as the other thing". Embedded in that mess of pronouns is something important, and I made sure to dig into this idea. After writing that student's words on the board, I pressed him to define "it" and "the other thing". Once he did, we got to the idea that f(x) was the repeated notation, and that if we're told that f(x)=37 and that f(x)=10+18x, then we can use the transitive property of equality to get the equation 37=10+18x, which isn't such a big deal to solve.

My other approach to the final problem is to write 37 in the output column of the table of values, and this again lends clarity to a lot of kids about what's going on here.

There are three versions of Part 1a of the project, Easy, Medium, and Hard, in the resources folder here. What differs between the three versions is how "nicely" the numbers work out in each scenario, and a few of the "Interpretation Questions" are different from one version to the next.

In addition to the mathematical goals of this project, students will also gain some financial literacy. I like to be able to start conversations by saying things like, "If you've ever wondered how to save up $1000, here's one way to make a plan" Kids realize that that's not easy to do, but this also makes it seem a little more achievable to some of them.

Ways to Interpret a Solution

Over the next few lessons, I'll spend a lot of time helping students distinguish between how they'd say something in a math context, and how people actually talk in real life. Compare, for example, this student's answer to question #2a to what this student writes in the same place. They've both done the math right. The former student simply takes an answer and writes it, while the latter uses the conversation tone that someone might actually use. So for the first student, my role is to ask whether or not she's ever heard anyone answer a question by saying 88.88 weeks, which is a fun conversation to have, and which will help her build an understanding of what it means to interpret a function, its parameters, or a solution to a problem. For the second student, I might want to touch on the idea of rounding up, but I can also see that he's got a pretty good grasp on what we're actually doing.

Today's work sets the stage for conversations of this kind for the next week or so.